Solving Systems Of Equations By Substitution Method

by ADMIN 52 views

In the realm of mathematics, systems of equations play a crucial role in modeling and solving real-world problems. These systems involve two or more equations with multiple variables, and the goal is to find the values of these variables that satisfy all equations simultaneously. One powerful method for solving systems of equations is the substitution method, which involves expressing one variable in terms of another and substituting this expression into the other equation.

Understanding the Substitution Method

The substitution method is a versatile technique that allows us to solve systems of equations by systematically eliminating variables. The core idea behind this method is to isolate one variable in one equation and then substitute the expression for that variable into the other equation. This substitution results in a single equation with one variable, which can be easily solved. Once we find the value of one variable, we can substitute it back into either of the original equations to find the value of the other variable.

The power of the substitution method lies in its ability to simplify complex systems of equations into manageable forms. By strategically substituting expressions, we can reduce the number of variables in an equation, making it easier to solve. This method is particularly effective when one equation can be easily solved for one variable in terms of the other.

Let's delve deeper into the steps involved in the substitution method:

  1. Isolate a variable: Begin by choosing one of the equations and isolating one variable. This means expressing one variable in terms of the other. Look for an equation where a variable has a coefficient of 1 or -1, as this will make the isolation process simpler.
  2. Substitute the expression: Once you've isolated a variable, substitute the expression you obtained into the other equation. This will result in an equation with only one variable.
  3. Solve for the remaining variable: Solve the equation you obtained in step 2 for the remaining variable. This will give you the numerical value of one of the variables.
  4. Substitute back to find the other variable: Substitute the value you found in step 3 back into either of the original equations or the expression you obtained in step 1. This will allow you to solve for the other variable.
  5. Check your solution: To ensure accuracy, substitute the values you found for both variables into both original equations. If both equations are satisfied, your solution is correct.

Applying the Substitution Method: A Step-by-Step Example

To illustrate the substitution method, let's consider the following system of equations:

egin{cases} 2x + 3y = 3 \ y = 8 - 3x \end{cases}

In this system, we have two equations with two variables, $x$ and $y$. Our goal is to find the values of $x$ and $y$ that satisfy both equations simultaneously.

Step 1: Isolate a variable

Looking at the equations, we notice that the second equation, $y = 8 - 3x$, is already solved for $y$. This makes it convenient to use this equation for substitution.

Step 2: Substitute the expression

Now, we substitute the expression for $y$ from the second equation into the first equation:

2x+3(8βˆ’3x)=32x + 3(8 - 3x) = 3

This substitution replaces $y$ in the first equation with the expression $8 - 3x$, resulting in an equation with only one variable, $x$.

Step 3: Solve for the remaining variable

Next, we solve the equation obtained in step 2 for $x$:

2x+24βˆ’9x=32x + 24 - 9x = 3

Combine like terms:

βˆ’7x+24=3-7x + 24 = 3

Subtract 24 from both sides:

βˆ’7x=βˆ’21-7x = -21

Divide both sides by -7:

x=3x = 3

Therefore, we find that the value of $x$ is 3.

Step 4: Substitute back to find the other variable

Now that we know the value of $x$, we can substitute it back into either of the original equations to find the value of $y$. Let's use the second equation, $y = 8 - 3x$:

y=8βˆ’3(3)y = 8 - 3(3)

y=8βˆ’9y = 8 - 9

y=βˆ’1y = -1

Thus, the value of $y$ is -1.

Step 5: Check your solution

To verify our solution, we substitute the values $x = 3$ and $y = -1$ into both original equations:

Equation 1: $2x + 3y = 3$

2(3)+3(βˆ’1)=6βˆ’3=32(3) + 3(-1) = 6 - 3 = 3

The first equation is satisfied.

Equation 2: $y = 8 - 3x$

βˆ’1=8βˆ’3(3)-1 = 8 - 3(3)

βˆ’1=8βˆ’9-1 = 8 - 9

βˆ’1=βˆ’1-1 = -1

The second equation is also satisfied.

Since both equations are satisfied, our solution $x = 3$ and $y = -1$ is correct.

Rewriting the Resulting Equation

In the given problem, the question asks for the resulting equation after substituting the value of $y$ from the second equation into the first equation. Following the steps above, we performed the substitution:

2x+3(8βˆ’3x)=32x + 3(8 - 3x) = 3

This is the resulting equation after the substitution. We can further simplify this equation, as we did in step 3, to solve for $x$. However, the question specifically asks for the equation immediately after the substitution, which is:

2x+3(8βˆ’3x)=32x + 3(8 - 3x) = 3

Advantages and Disadvantages of the Substitution Method

The substitution method offers several advantages:

  • Simplicity: It's a relatively straightforward method to understand and apply, especially for systems where one variable is already isolated or can be easily isolated.
  • Versatility: It can be used to solve a wide range of systems of equations, including those with linear and nonlinear equations.
  • Efficiency: In some cases, it can be more efficient than other methods, such as elimination, particularly when one variable is already isolated.

However, the substitution method also has some disadvantages:

  • Complexity: For systems where no variable is easily isolated, the substitution method can become more complex and time-consuming.
  • Error-prone: There's a higher chance of making algebraic errors during the substitution and simplification steps.
  • Not always the best choice: For certain systems, other methods, like elimination, might be more efficient.

When to Use the Substitution Method

The substitution method is most effectively used when:

  • One of the equations is already solved for one variable in terms of the other.
  • One of the variables has a coefficient of 1 or -1 in one of the equations, making it easy to isolate.
  • The system involves a mix of linear and nonlinear equations.

In situations where these conditions are not met, other methods, such as elimination or matrix methods, might be more suitable.

Common Mistakes to Avoid

When using the substitution method, it's essential to be mindful of potential errors:

  • Incorrect substitution: Ensure you substitute the expression for the correct variable into the correct equation.
  • Distributing negatives: When substituting an expression with a negative sign, remember to distribute the negative sign to all terms inside the parentheses.
  • Algebraic errors: Be careful when simplifying the equation after substitution, as algebraic errors can easily occur.
  • Forgetting to solve for both variables: After finding the value of one variable, remember to substitute it back to find the value of the other variable.
  • Not checking the solution: Always check your solution by substituting the values into the original equations to ensure they are satisfied.

Conclusion

The substitution method is a valuable tool for solving systems of equations. By strategically isolating variables and substituting expressions, we can simplify complex systems and find solutions efficiently. Understanding the steps involved, the advantages and disadvantages, and common mistakes to avoid will empower you to confidently apply the substitution method in various mathematical contexts.

Remember, practice is key to mastering any mathematical technique. Work through various examples, and you'll become proficient in using the substitution method to solve systems of equations.

What is the equation that results from substituting the expression for $y$ from the second equation, $y = 8 - 3x$, into the first equation, $2x + 3y = 3$?

Solving Systems of Equations by Substitution Method